$ 0.\overline{3} \div 3.\overline{81} = {?} $
Answer: First convert the repeating decimals to fractions. $\begin{align*} 10x &= 3.3333...\\ x &= 0.3333...\end{align*} $ $\begin{align*} 9x &= 3 \\ x &= \dfrac{3}{9}\end{align*} $ $\begin{align*} 100y &= 381.8181...\\ y &= 3.8181...\end{align*} $ $\begin{align*} 99y &= 378 \\ y &= \dfrac{378}{99}\end{align*} $ So, the problem becomes: $ \dfrac{3}{9} \div \dfrac{378}{99} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{3}{9} \times \dfrac{99}{378} = {?} $ $ \phantom{\dfrac{3}{9} \times \dfrac{378}{99}} = \dfrac{3 \times 99}{9 \times 378} $ $ \phantom{\dfrac{3}{9} \times \dfrac{378}{99}} = \dfrac{3 \times \cancel{99}11} {\cancel{9} \times 378} $ $ \phantom{\dfrac{3}{9} \times \dfrac{378}{99}} = \dfrac{33}{378} $ Simplify: ${= \dfrac{11}{126}}$